Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems

Abstract: This work provides a geometric approach to the study of bifurcation and rate induced transitions in a class of non-autonomous systems referred to herein as $\textit{asymptotically slow-fast systems}$, which may be viewed as 'intermediate' between the (smaller resp. larger) classes of asymptotically autonomous and non-autonomous systems. After showing that the relevant systems can be viewed as singular perturbations of a limiting system with a discontinuity in time, we develop an analytical framework for their analysis based on geometric blow-up techniques. We then provide sufficient conditions for the occurrence of bifurcation and rate induced transitions in low dimensions, as well as sufficient conditions for 'tracking' in arbitrary (finite) dimensions, i.e. the persistence of an attracting and normally hyperbolic manifold through the transitionary regime. The proofs rely on geometric blow-up, a variant of the Melnikov method which applies on non-compact domains, and general invariant manifold theory. The formalism is applicable in arbitrary (finite) dimensions, and for systems with forward and backward attractors characterised by non-trivial (i.e. non-constant) dependence on time. The results are demonstrated for low-dimensional applications.